434f Prof. Sylvester on Rational Derivation, 



term by the leading term of that anormal prime of the same 

 degree which has the lowest dimensions, — that finds its way 

 into the numerator. The rest of the formation remaining un- 

 disturbed, uiiless and until a new failure have taken place. 



Note on SturtrCs Theorem. 



When one of the equations of coexistence is the differential 

 coefficient with respect to the repeated term of the other, 

 the prime derivatives given in theorem (2.) which coincide 

 in this case with Sturm's auxiliary functions reduced to their 

 lowest terms, may be exhibited under an integral aspect. 



Let SPD intimate that the squared product of the differ- 

 ences is to be taken of the quantities which follow it. 



Let Si indicate the sum of the quantities to which it is pre- 

 fixed. 



Sg the sum of the binary products. 



Sg the sum of the ternary products, and so on. 



Let hi h^ ... hn be the roots of any equation. 



Then Sturm's last auxiliary function may be replaced by 

 SPD {k, h^ ... h„). 



The last but one may be replaced by 

 XSFD{h,h^...hn-i)x + tS,{h^hs...hn-i)SFD{h,h^...h^--'). 



The one preceding by 

 tSVD{h, h^...hn-2)x^+2S,{h,h^...hn-2)SFB{h,h^.,.7i„-^)x 

 + 2 83 (^1 ho ... hn-2) SPD (/^i /zg ... hn-2) and so on. 



Thus then Sturm's rule for determining the absolute number 

 of real roots in an equation is based wholly and solely upon 

 the following 



ALGEBRAICAL PROPOSITION. 



If there be n quantities, real and imaginary, the imaginary 

 ones entering in pairs, as many changes of sign as there are 

 in the terms 



2 SPD {h, h^) 



t SPD [h^ 7/2 ... h.^) 



so many in number are these 

 pairs. 



S SPD (/?, h^ ... hn-^) 

 SPD [h, h^ ... hn) _^ 



Query (L) Is there no proposition applicable to any («) 

 quantities iiohate'cer ? 



Query (2.) Is there no faintly analogous proposition ap- 

 plicable to higher powers than the squares ? 



Query (3.) Seeing that in forming the coefficients in the 

 equation to the squares of the differences, we pass from (w) 



functions oftherootS; to w-^— and not 7/ functions, of their 



